Planar and oblique shock wave interaction with a droplet seeded gas cylinder

Transcription

1 University of New Mexico UNM Digital Repository Mechanical Engineering ETDs Engineering ETDs Planar and oblique shock wave interaction with a droplet seeded gas cylinder Evan Johnson Follow this and additional works at: Recommended Citation Johnson, Evan. "Planar and oblique shock wave interaction with a droplet seeded gas cylinder." (2010). This Thesis is brought to you for free and open access by the Engineering ETDs at UNM Digital Repository. It has been accepted for inclusion in Mechanical Engineering ETDs by an authorized administrator of UNM Digital Repository. For more information, please contact disc@unm.edu.

2

3 Planar and oblique shock wave interaction with a droplet seeded gas cylinder by Evan PR Johnson B.S., Mechanical Engineering, University of New Mexico, 2007 THESIS Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science Mechanical Engineering The University of New Mexico Albuquerque, New Mexico December, 2009

4 c 2009, Evan PR Johnson iii

5 Acknowledgments I would like to acknowledge all of those who encouraged me, assisted me, and inspired me during the writing of my Master s thesis. Without the help from God, family, friends, and colleagues none of this would have been possible. A special thanks to Riccardo Bonazza for creating the online W ist L gas dynamics calculator that assisted in checking calculations for my thesis. iv

6 Planar and oblique shock wave interaction with a droplet seeded gas cylinder by Evan PR Johnson ABSTRACT OF THESIS Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science Mechanical Engineering The University of New Mexico Albuquerque, New Mexico December, 2009

7 Planar and oblique shock wave interaction with a droplet seeded gas cylinder by Evan PR Johnson B.S., Mechanical Engineering, University of New Mexico, 2007 M.S., Mechanical Engineering, University of New Mexico, 2009 Abstract We present an experimental study of a shock interaction with an initially diffuse heavy-gas cylinder seeded with submicron-scale glycol droplets. Unlike most earlier studies, the investigation covers not just a quasi-two-dimensional geometry, where the axis of the cylinder is parallel to the plane of the shock, but also the oblique interaction at an angle of 15 between the cylinder axis and the plane of the shock wave. Our experimental data cover the range of Mach numbers from 1.2 to 2.4. The heavy gas cylinder is produced by injecting sulfur hexafluoride pre-mixed with glycol vapor into the test section of a tiltable shock tube through a co-flowing nozzle, with the gravity-driven flow of the heavy gas stabilized by an annular flow of air in the downward direction. Droplets in the gas cylinder are visualized via Mie scattering of diffuse white light. Two views of the flow side and top are simultaneously vi

8 captured by a high-speed gated and intensified CCD camera, producing a spatially and temporally resolved description of the evolution of the gas cylinder upon impulsive acceleration. While the observations for the planar interaction reveal that the large-scale flow structure remains largely two-dimensional, confirming the assumptions of earlier studies, during the oblique shock interaction, we observe evidence of flow evolution in three dimensions, including asymmetric interaction of the gas cylinder with the boundary layers forming on the walls of the shock tube, and rotation of this cylinder in the vertical plane parallel to the streamwise direction. vii

9 Contents List of Figures x Glossary xiv 1 Introduction Governing Equations Shock-driven evolution of gaseous interfaces: earlier work Experimental Setup and Diagnostics Overview Injection System Diagnostics Additional Capabilities Acoustic Pre-shock Results of Experiments Horizontal Experiments viii

10 Contents 3.2 Oblique Experiments Observations Conclusions and Future Work Conclusions Future Work References 44 ix

11 List of Figures 1.1 Simplified representation of a planar shock acceleration of an initially diffuse cylindrical density interface: left - before the acceleration, right - after the acceleration. The initial misalignment between gradients of pressure (black arrows) and density (grey arrows) leads to deposition of vorticity (white dashed arrows). After Palekar et al. [5] Images for a M=1.2 shock interacting with an f/s interface where the density ratio is equal to 3 and interface-shock plane angle theta is equal to 30, after Yang et al. [8]. Presented is a time sequence: density (a1)-(a4) left column; pressure (b1) and (b2), center column; and vorticity (c1)-(c4), right column. The times for these images are row 1, t = (100); row 2, t = (200); row 3, t = (800), and row 4, t = (1400). Direction of the flow is from left to right. From Yang et al. [8] Schematic of physical domain and parameters in shock accelerated density stratified interfaces. (a) Planar interface; (b) sinusoidally perturbed interface; and (c) circular interface. From Samtaney and Zabusky [9] x

12 List of Figures 1.4 Circulation per unit original length for M=1.05 and M=2.0. The circles, diamonds, and squares represent the exact circulation, approximate circulation, and filled numerical circulation, respectively, Samtaney and Zabusky [9] Circulation for sinusoidal interface and the U-side of the circular bubble for density ratios of 1.2, 3.0, 6.0, and Open squares indicate the sinusoidal interface with amplitude of 10 and wavelength of 80. Filled squares indicate the sinusoidal interface with amplitude of 40 and wavelength of 80. Circles indicate the circular bubble. Solid line represents the analytical result. From Samtaney and Zabusky [9] A sequence of PLIF images (diagonal slices) along with corresponding concentration maps obtained from the simulation. From Krivets et al. [12] Diagram showing the layout of the shock tube and the individual sections Diagram showing the injection system with co-flow implemented for the planar shock wave experiments Diagram showing the injection system with co-flow implemented for the oblique shock wave experiments DRS Imaging Imacon 200 high-speed camera [19] Schematic showing how the camera, delay generator, lasers, and transducers are configured. For details refer to Chavez et al. [20] Diagram showing the orientation of Imacon 200 relative to test section (left) and an individual frame representative of typical experimental results (right) xi

13 List of Figures 2.7 Raw frames without acoustic pre-shock Raw frames with acoustic pre-shock Processed image sequence, inverted and contrast-enhanced, Mach 2.4 acceleration of SF 6 gas cylinder with shock tube in the horizontal orientation Translation of the center of mass of the gas cylinder from t = 0 ms to t = ms for Mach numbers 1.2, 1.9, and 2.4 with shock tube in horizontal orientation Width (in the direction normal to the streamwise) of counter-rotating vortex pair versus time with shock tube in the horizontal orientation Span (in the streamwise direction) of counter-rotating vortex pair versus time with shock tube in the horizontal orientation Processed image sequence, inverted and contrast-enhanced, Mach 2.4 acceleration of SF 6 gas cylinder with shock tube inclined 15 from horizontal Translation of the center of mass of the gas cylinder from t = 0 ms to t = ms for Mach numbers 1.2, 1.9, and 2.4 with shock tube in the inclined orientation Change in the angle of the gas cylinder relative to the shock front versus time Width (in the direction normal to the streamwise) of the counterrotating vortex pair versus time with shock tube in the inclined orientation xii

14 List of Figures 3.9 Span (in the direction normal to the streamwise) of the counterrotating vortex pair versus time with shock tube in the inclined orientation Piston velocity versus Mach number xiii

15 Glossary ρ 1 Density before the shock wave ρ 2 Density after the shock wave γ Heat capacity ratio M 1 Mach number before the shock wave M 2 Mach number after the shock wave p 1 Pressure before the shock wave p 2 Pressure after the shock wave v 1 Specific volume before the shock wave v 2 Specific volume after the shock wave T 1 Temperature before the shock wave T 2 Temperature after the shock wave xiv

16 Chapter 1 Introduction Richtmyer-Meshkov Instability (RMI) occurs when an interface between fluids of contrasting densities is impulsively accelerated. One method of impulsive acceleration is the passage of a shock wave through the fluid interface. The development of the instability begins with perturbations that are small in amplitude which initially grow linearly with time. This is followed by a nonlinear regime with bubbles appearing on the side of a light fluid penetrating a heavy fluid, and with spikes appearing on the side of a heavy fluid penetrating a light fluid. A turbulent regime eventually is reached and the two fluids mix. In our experiments we see the RMI, but with some added complexities due to the presence of liquid droplets. Modeling a compressible multiphase flow with mixing gases and particle/droplet phases is relevant to our work in addition to other applications. Some of these applications include modeling conditions inside of internal combustion engines, rocket engines, and inertial confinement fusion. Our work will benchmark different phenomena that occur when a shock wave accelerates a gas cylinder that is at an oblique angle to the shock front itself. Our initial conditions are three-dimensional, not nominally two-dimensional like most earlier experiments that have been done. Multiple complex physical phenomena have to be taken into consideration which occur when an oblique shock wave interacts with a density interface. Several insta- 1

17 Chapter 1. Introduction bilities are introduced which interfere with one another, adding more complexity to the interaction. If a constant normal acceleration is present it feeds energy to the Rayleigh-Taylor (RT) instability or, depending on the sign of ga (g=acceleration, A=Atwood number), it acts to stabilize the KH and RM instabilities [1]. In this thesis, I will present our experimental/diagnostic set up, results of our experiments, as well as observations and future planned experiments. 1.1 Governing Equations The following equations describe the conditions before and after the shock wave. We can use them to determine certain quantities and compare them to our experimental results. From one side of the shock wave to the other, there are significant differences in pressure, density, and temperature among other properties. The following equations are fundamental to our study of shock waves and particle interactions. They describe the conditions before (subscript 1) and after (subscript 2) the shock. In this section, the nomenclature and equations follow Landau and Lifshitz [2]. The density change is direct result of gas compression [2]: ρ 2 ρ 1 = (γ + 1) M 2 1 (γ 1) M (1.1) The pressure jump in the gas leads to compression of the particulate phase [3]: p 2 p 1 = 2γ γ + 1 M 1 2 γ 1 γ + 1 (1.2) The temperature jump in gas leads to the heating of the specimen under test, e.g.. particles, droplets, etc. [2]: T 2 = [2γM 1 2 (γ 1)] [(γ 1) M ] T 1 (γ + 1) 2 M1 2 (1.3) 2

18 Chapter 1. Introduction The Mach number M 2 is given in terms of M 1 [2] as M 2 2 = 2 + (γ 1) M 2 1 2γM 2 1 (γ 1) (1.4) We can give limiting results for very strong shock waves, in which (γ 1) p 2 is very large compared with (γ + 1) p 1 [2]: ρ 1 = γ 1 ρ 2 γ + 1 (1.5) T 2 T 1 = (γ 1) p 2 (γ + 1) p 1 (1.6) The ratio T 2 /T 1 increases to infinity as p 2 /p 1 increases [2]. Thus, the temperature discontinuity in a shock wave, like the pressure discontinuity, can be arbitrarily high. The density ratio tends to a constant limit. For a monatomic gas the limit is ρ 2 =4ρ 1 [2]. While these equations are sufficient for single phase gaseous flows, the addition of particles or droplets changes the flow physics, especially during the transition to turbulence. There are many applications where the dispersion of particles in turbulent shear flows is of great interest and importance. The injection of gasoline in an internal combustion engine, for example, is a very important application. Some others include particle spray in rocket engines, flow of concrete in transportation pipes, and pollutants in the air or bodies of water. Depending on the mass loading of particles that exist in a particular flow, they can actually modify the turbulence structure in the carrier fluid [4]. In our shock tube experiments, the injection of a Sulfur Hexafluoride (SF 6 ) gas cylinder seeded with glycol droplets qualifies as a particle laden flow. Understanding the influence of the particles on the turbulence structures could prove extremely useful in characterizing certain observed flow phenomena. Elghobashi [4] realized that the prediction of these transport phenomena requires knowledge of the two-way nonlinear coupling between the particles and turbulence, 3

19 Chapter 1. Introduction i.e. the response of the discrete particles to the turbulent motion of the fluid, and the effect of the particles motion on the frequency spectrum of turbulence [4]. Elghobashi used a direct numerical simulation (DNS) to predict turbulent flows laden with solid particles. He defined several quantities that characterize the effects that the particles will have on the flow. Depending on the particle size, concentration, and spacing between one another, the particles have different effects on the flow. One-waycoupling, two-way-coupling, or four-way-coupling are terms that were used to designate the interaction between the particles and turbulence. In one-way-coupling, the particle dispersion depends on the state of turbulence but there is no feedback to the turbulence itself. Two-way-coupling is second regime where the particle loading is large enough to alter the turbulence structure. In this regime there is an increased dissipation rate of turbulence energy as the diameter of the particle decreases for the same particle material and fluid viscosity. However, as the particle response time increases for a given particle volume fraction, the particle Reynolds number increases and vortex shedding takes place resulting in enhanced production of turbulence energy [4]. Flows that fall into one and two-way coupling are often referred to as dilute suspensions. In four-way-coupling, the particle loading is large enough that there are actually particle/particle collisions that take place, hence the term four-waycoupling. Due to the particle loading in these flows, they are often referred to as dense suspensions. The flows are very complex in this regime, so most studies are conducted using dilute suspensions [4]. DNS provides a modeling-free, three-dimensional, instantaneous velocity field for the fluid in simple turbulent flows. This velocity field can be used to calculate the three-dimensional trajectory of a particle from which the dispersion statistics can be obtained [4]. This work showed that depending on the particle characteristics, mass loading in the flow, and the carrier fluid, particles can actually alter the turbulence structure in a shear flow. However, at the present state of the art, while DNS can be used to understand local particle dynamics, spatially resolved DNS of a multiphase 4

20 Chapter 1. Introduction mixing flow on large scales remains prohibitively computationally expensive, thus requiring different models (such as large eddy simulation, or LES), which have closure assumptions necessitating benchmarking with experimental results. The following section focuses on the Richtmyer-Meshkov instability when the particulate phase is either not considered, not present, or is present as a tracer (in one-way coupling regime). Experiments related to turbulent mixing and baroclinic vorticity generation are also discussed. Since all of the aforementioned topics relate to our shock tube experiments, a good understanding of these topics will help us to interpret the results and observations of our experiments. 1.2 Shock-driven evolution of gaseous interfaces: earlier work As discussed earlier, the RMI arises when a density gradient in a fluid (gas) is subjected to an impulsive acceleration (e.g., due to a shock wave passage). The evolution of RMI is non-linear and hydrodynamically complex and hence is a very good test problem to validate numerical codes. In a paper by Palekar et al. [5], a two-dimensional numerical simulation of RMI-driven evolution of the flow produced by shock acceleration of a diffuse heavy gaseous cylinder embedded in lighter gas is presented. Comparison of the late-time flow statistics between experiment and numerics elucidated the limitations inherently present in a two-dimensional simulation of a spatially three-dimensional flow, even if the large-scale flow structure is nominally two-dimensional [5]. This section focuses on the Richtmyer-Meshkov instability when the particulate phase is either not considered, not present, or is present as a tracer (in one-way coupling regime). This interfacial instability was theoretically predicted by a Los Alamos theorist Richtmyer (1960) [6], and first experimentally observed by a Russian exper- 5

21 Chapter 1. Introduction Figure 1.1: Simplified representation of a planar shock acceleration of an initially diffuse cylindrical density interface: left - before the acceleration, right - after the acceleration. The initial misalignment between gradients of pressure (black arrows) and density (grey arrows) leads to deposition of vorticity (white dashed arrows). After Palekar et al. [5]. ' imentalist working in VNIIEF (Sarov), Meshkov (1969) [7]. The fluid configuration leading to a frequently considered RMI problem is shown in Fig Two fluids with different properties are separated by an initially diffuse interface. The shock travels from light fluid (gray color) to heavy fluid (black color) through the interface. Summarizing previous studies, Palekar et al. [5] described the development of RMI in this configuration as following a three-stage process of evolution. the short initial stage is linear. In this stage, the incident shock wave collides with a perturbed material interface and bifurcates into a transmitted shock and a reflected wave. In Fig. 1.1, left, the pressure and density gradients are locally misaligned. This misalignment leads to baroclinic generation of vorticity at the interface. The resulting vortex roll-up (Fig. 1.1, right) leads to growth of the perturbation amplitude of the interface. The flow-field in this stage is deterministic. Later in this stage, spikes and bubbles appear on the interface. A bubble is a portion of the light fluid penetrating 6

22 Chapter 1. Introduction into the heavy fluid and a spike is a portion of the heavy fluid penetrating into the light fluid. The second stage is non-linear deterministic. In this stage the spikes and bubbles grow substantially. The amplitude of perturbation grows to the order of the wavelength and hence now the flow is non-linear. In the later part of this stage, roll-up of material into the vortex cores occurs. Roughly at the same time, small scales also appear in the flow. In the third stage, the secondary instabilities become pronounced and lead to the onset of turbulence with chaotic mixing. Being turbulent in nature, this stage is dominated by three-dimensional physics, unlike the first two stages [5]. The classical Rayleigh-Taylor and Kelvin-Helmholtz instabilities refer to the exponential growth of interface perturbations under the action of a constant acceleration and under the action of shear, respectively. A shock normal to the interface induces an impulsive acceleration and triggers the Richtmyer-Meshkov instability which grows linearly in time, provided there is no post-shock acceleration or shear. Mikaelian [1] derived analytical expressions using a generalization of Richtmyer s technique. This meant treating the shock as an instantaneous acceleration of incompressible fluids. This was mainly to avoid the complexity of the fully compressible problem. As mentioned earlier, the problem of an oblique shock on a plane interface is well known, and the problem of a normal shock on a perturbed interface (the Richtmyer-Meshkov problem) is also quite familiar, the combined problem of an oblique shock on a perturbed interface remains unsolved [1]. Mikaelian considered two fluids having different densities and horizontal directed velocities, resulting from the induced perturbation. The fluid interface lies in the x z plane. Through several complex mathematical expressions involving densities, velocities, wave numbers, and an expression for the perturbation as a function of time, Mikaelian was able to describe the evolution of the perturbation at an interface after an oblique shock strike changed the normal velocity and parallel velocities. Both instantaneous as well as subsequent steady evolution expressions were derived. Some applications for this work were in inertial-confinement-fusion (ICF) capsules. 7

23 Chapter 1. Introduction Due to the amplification of the perturbation upon re-shock (when the shock is reflected back through the interface) it was proposed that this amplification was a possible mechanism for mixing in ICF capsules. In a typical ICF capsule, the fuel is a solid DT (Deuterium-Tritium) shell surrounding a very low-density sphere of gaseous DT, the spark plug. At this interface the density contrast is usually very large, thus making the RM instability dominant [1]. Mikaelian noticed that unless the first shock is immediately followed by acceleration the KH term, which is always positive, would dominate. He concluded that the interplay between RT, KH, and RM instabilities is complicated by the fact that they all have different dependence on different wave numbers (relating to the induced interface perturbations) and the compressible situation is obviously much more complex, even in the completely linear regime. Yang et al. [8] studied the results of direct numerical simulations of inviscid planar shock-accelerated density stratified interfaces in two dimensions and compared them to previously conducted shock tube experiments. Heavy-to-light (slow/fast or s/f) and light-to-heavy (fast/slow, or f/s) gas interfaces were examined and earlytime impulsive vorticity deposition and the evolution of coherent vortex structures were emphasized and quantified [8]. They determined that a second-order numerical Godunov scheme yielded excellent agreement with shock-polar analysis at early time. A more physical vortex interpretation explains the commonly used (i.e. linear paradigm) designations of unstable and stable for the f/s and s/f interfaces, respectively. In this work, there are some correlations to the work by Mikaelian in that they both are interested in RT instabilities. However Yang et al. were more interested in describing the RT instability in terms of a vortex layer (large-scale translation and rotation), asymmetric tip vortex roll-up and binding, layer instability, convective mixing, and baroclinic vorticity generation from secondary shock-interface interactions [8]. Yang et al. were particularly interested in the early-time stability and the longerterm evolution of a shock-accelerated interface. Yang et al. considered two cases. 8

24 Chapter 1. Introduction Both were characterized by a shock wave propagating through a fluid of one density, striking a contact interface, and passing into a region of a different density, where the two regions are initially in pressure balance. They defined the setup in terms of sound speed or wave impedance for a physically meaningful reason. The results of a linear stability analysis showed that the s/f interaction is stable and the f/s interaction is unstable. Whether an interaction is considered stable or unstable is relative to the length of the term being considered, which if it is too large, the surfaces of both interactions become increasingly distorted (unstable). Many analytical efforts have been confined to early time event or small infinitesimal perturbations so that advantage can be taken of linearity. However, beyond the early time events and small infinitesimal perturbations, both s/f and f/s interfaces depart drastically from their initial configurations. When an interface is deemed either stable or unstable, this can be misleading since it is only deemed such for a very short period of time. Collision of two vortex layers of opposite signs was used to model this. Yang et al. focused mainly on the primary and secondary features for the shock-single-interface interaction as well as demonstrating that the initial deposition of vorticity can be describe quantitatively by a shock-polar analysis (SPA), when the latter provides solutions [8]. Yang et al. [8] used the conservative form of Euler equations in two dimensions, namely the mass, momentum, and energy conservation equations for an inviscid, compressible fluid to model the shock tube experiments. Initially they separated the two gases of different densities by a sharp interface. In experiment, this would correspond to a membrane or some similar gas separation method. By contrast, in our experiments described in the following chapters, the density interface is gravitystabilized and initially diffuse. The interface is oriented so that it lies at a 15 angle from the vertical (Fig. 1.2). While Fig. 1.2 refers to M = 1.2, studies have also been performed with increased Mach number and incident shock angle. As either the Mach number or incident shock 9

25 Chapter 1. Introduction Figure 1.2: Images for a M=1.2 shock interacting with an f/s interface where the density ratio is equal to 3 and interface-shock plane angle theta is equal to 30, after Yang et al. [8]. Presented is a time sequence: density (a1)-(a4) left column; pressure (b1) and (b2), center column; and vorticity (c1)-(c4), right column. The times for these images are row 1, t = (100); row 2, t = (200); row 3, t = (800), and row 4, t = (1400). Direction of the flow is from left to right. From Yang et al. [8]. angle are increased for a f/s interaction, the processes are stronger and more rapid. To simulate the flow field, Yang et al. used a second-order Godunov scheme which is very suitable for problems involving complicated nonlinear wave interactions [8]. At a later time once the shock has propagated, the interface rotates away from the wall of the tube and it is evident that there is breaking of symmetry, since the location of the upper layer tip is farther from the wall than the lower tip. The difference is mainly due to the interaction of the interface with the following two positive weak vortex sheets of different strengths: one is at the lower boundary; the other is at the upper boundary. These are caused by bent transmitted and bent shock waves respectively. Additionally, these shocks also emit a reflected shock 10

26 Chapter 1. Introduction interacting with the interface. The changes in pressure due to these phenomena are relatively weak. The interface is essentially a shear layer, and if it is sufficiently strong, vortical rollers will develop as a result of the nonlinear growth of the KH instability. Note that interface stretching is a stabilizing phenomenon postponing the appearance of rollers [8]. Several different configurations were used including various fluids such as air, helium, and R22 refrigerant (Freon) in different configurations to obtain either a f/s or s/f interface. Yang et al. demonstrated the importance of shock-polar analysis and a vorticity interpretation of physical trends. They found a good agreement between their numerical simulations and experimental investigations. Further studies by Samtaney and Zabusky [9] were conducted on the shock polar analysis and analytical expressions for vorticity deposition in shock-accelerated density-stratified interfaces. Vorticity is deposited due to baroclinic effects on the surface of a density-stratified interface accelerated by a shock. An analytical expression was presented, derived from shock polar analysis, for circulation per unit length on a fast-slow planar density interface inclined at an angle to the incident shock [9]. Samtaney and Zabusky used an integration of their analytical expression to yield total circulation on non-planar interfaces (sinusoidal and semicircular interfaces) accelerated by shocks. Their results correlated strongly with the diagnostics they obtained from numerical experiments using a second-order Godunov code for the Euler equations. Elemental processes that occur between a shock wave and a density inhomogeneity are of practical interest in combustion and inertial confinement (laser driven) fusion. Of special interest is the early-time stability and growth rate for shockaccelerated interfaces, which may be explained from the point of view of vortex dynamics [9]. Samtaney and Zabusky believed that accurate quantification of vorticity generation is a very important goal. In this particular experiment they presented analytical expressions for circulation at fast-slow planar gas interfaces using shock polar analysis. With those results they predicted analytically the circulation on nonplanar interfaces (sinusoidally perturbed interfaces and bubbles), see Fig They 11

27 Chapter 1. Introduction found a strong agreement between their analytical results and their numerical experiments. Samtaney and Zabusky used problem settings similar to those of Yang et al. A rectangular shock tube with a shock wave propagating through it into a gas of density 1 and an interface separating a gas of density 2 was modeled. Three different possible physical situations were a planar interface inclined at an angle α; a sinusoidally perturbed vertical interface with amplitude A and wavelength λ; or a bubble of radius r 0. Effects of viscosity and changes in specific heat ratio were omitted, as they played a minor role in this parameter domain. As in the simulations by Yang et al., there were two generic classes of interactions that exist. One in which the shock crosses into a fluid with a higher sound speed (i.e. lower density, referred to as the slow/fast or s/f interaction) and the case where the shock passes into a region with a lower sound speed (i.e. higher density, referred to as the fast/slow or f/s interaction). They, however, only considered the f/s interaction for this particular set of simulations (Fig. 1.3). In shock-polar analysis (SPA) one assumes a frame of reference, which is stationary, with respect to the node where all the shocks meet [9]. They assumed also that initially, both gases are perfect, inviscid, and of constant identical specific heat ratio (γ). Through using analytical expressions that are functions of parameters such as incident (where angle of incidence is α), reflected, and transmitted shocks, streamlines, deflections of shocks, pressures, and Mach numbers, they then derived the SPA equation, which was used to obtain the analytical results. After showing the normalized circulation per unit of length of the interface and the numerical result for a density ratio of 3, Mach number of 1.05 and 2.0, it was apparent that the SPA result terminates at a certain critical angle while the approximate result can be extended to α = π/2, see Fig. 1.4 [9]. At a large α, the agreement was better for moderate to large Mach numbers. Similar observations were made for large density ratios (Fig. 1.5). A significant source of error arises from comparing the SPA vortex sheet results with an evolving vortex that spreads because of numerical diffusion [9]. This means that for a larger α, the error is expected to be larger as well. 12

28 Chapter 1. Introduction w J/o UO Figure 1.3: Schematic of physical domain and parameters in shock accelerated density stratified interfaces. (a) Planar interface; (b) sinusoidally perturbed interface; and (c) circular interface. From Samtaney and Zabusky [9]. From this work, exact and approximate analytical expressions for circulation deposited on a fast-slow interface accelerated by a shock were presented. Analytical expressions were obtained for vorticity deposition on sinusoidally perturbed and circular interfaces that are in good agreement with numerical results [9]. Samtaney and Zabusky intended to continue work and show where changes in the ratio of the specific heats of the two gases across the interface do not significantly effect the circulation of vorticity. Additionally, they used analytical expressions to develop models to predict the vorticity deposition in shock-bubble interactions and for slow-fast interfaces. For sinusoidally perturbed interfaces, they used the expression for circulation and related it to the growth rate of perturbations in the RMI environment [9]. 13

29 Chapter 1. Introduction Anaie. a Figure 1.4: Circulation per unit original length for M=1.05 and M=2.0. The circles, diamonds, and squares represent the exact circulation, approximate circulation, and filled numerical circulation, respectively, Samtaney and Zabusky [9]. Studies have been performed on the effects of droplets interacting with oblique shock waves. The main point of one study in particular was to show the breaking-up effects of droplets when they passed through a straight oblique shock wave. In a study by Utheza et al. [10], it was expected that by the addition of particles that the initial slope of the shock wave would change and in some cases that the shock would turn into a compression zone. They noted that among many numerical gas-particles flow simulations proposed in literature, the two fluids model had often been retained. This model said that both the gaseous and dispersed phases are each considered to be a distinct continuous medium. Through the particle surface, momentum and energy were exchanged through viscosity and the thermal conductivity [10]. Other researchers gave distinct mechanisms for the fragmentation of a liquid droplet in gas flow [11]. In the study by Utheza et al. [10], it was assumed that the flow was unsteady and 14

30 Chapter 1. Introduction 1.25 E Figure 1.5: Circulation for sinusoidal interface and the U-side of the circular bubble for density ratios of 1.2, 3.0, 6.0, and Open squares indicate the sinusoidal interface with amplitude of 10 and wavelength of 80. Filled squares indicate the sinusoidal interface with amplitude of 40 and wavelength of 80. Circles indicate the circular bubble. Solid line represents the analytical result. From Samtaney and Zabusky [9]. two-dimensional, the compressible phase was exclusively constituted of steam and followed the ideal gas law, real fluid effects were taken into account only at liquid vapor interfaces, droplets were spherical and incompressible, at a given space point droplets had the same size, speed, and temperature, droplet-droplet interactions were not considered, and the volume occupied by droplets was small. The influence of a two-phase oncoming flow around a wedge where an attached stationary shock wave has developed was studied. The authors of this study investigated mainly the process of particle break-up and its effects on the oblique and straight initial shock. For t < 0, the flow around the wedge was a one-phase flow (water steam only). For each case the initial conditions of the two dimensional one phase flow were such that an attached shock wave exists at the tip of the wedge. At t = 0, the dispersed phase (droplets) was introduced upstream of the initial location of the stationary shock wave. The 15

31 Chapter 1. Introduction two-phase flow consisted of steam (gaseous phase) and water droplets (dispersed phase) which were assumed to be spherical, mono-dispersed. Upon modeling the flow mathematically and verifying the results numerically, they were able to simulate a gas-particle flow passing through a straight oblique shock wave. Their analysis allowed them to emphasize a certain number of original and noticeable results. These results included that generally, introducing particles in the flow weakens the shock wave. This was accentuated by droplet break-up occurring across the shock wave. The shock wave was weaker when mass and heat transfers were considered in addition to drag force. Also, the addition of the droplet break-up process accentuated the weakening of the shock wave. Particles became smaller after break-up if only the drag forces were taken into account [10]. This work could prove useful in our later studies and attempts to explain and understand the different phenomena that occur when an oblique shock wave interacts with a droplet-seeded gas cylinder. Since our gas cylinder is seeded with small glycol droplets which can be assumed to be spherical, close correlations can be made to give reason as to why certain phenomena are occurring. For example, in the oblique experiments section of Chapter 3 of this thesis, it is mentioned that some of the droplet-seeded gas cylinder material is advected into the boundary layer of the test section in our shock tube. Reasons for this could be very closely related to those verified by Utheza et al. Moreover, a slight bow develops in the gas cylinder as well, possibly due to drag forces or boundary layer interaction with the gas cylinder itself. Krivets et al. [12] used a vertical shock tube to study the single-mode threedimensional RMI. The interface was formed using opposed flows of air and SF 6 and the perturbation was created by the periodic motion of the gases within the shock tube. They used Planar laser induced fluorescence for flow visualization. Then they compared their experimental results which were obtained with a shock Mach number of 1.2 with a three-dimensional numerical simulation [12]. A code called Raptor, a multi-dimensional Eulerian adaptive mesh refinement 16

32 Chapter 1. Introduction (AMR) code, was used for the three-dimensional computations. It was developed and is currently supported by AX-Division at Lawrence Livermore National Laboratory (LLNL) [12]. There was a strong correlation between their experimental and numerical results, see Fig Figure 1.6: A sequence of PLIF images (diagonal slices) along with corresponding concentration maps obtained from the simulation. From Krivets et al. [12]. Since the resolution of that simulation was relatively low, the same degree of vortex roll-up was not quite achieved. However, the amplitude measurements showed excellent agreement between experiment and simulation, which illustrated a relative insensitivity of amplitude measurements to fine interface features [12]. Krivets et al. also noted that there is a dependence on the Atwood number to produce asymmetry in RM flows. 17

33 Chapter 1. Introduction Spatially and temporally resolved experimental RMI studies were carried out by Rightley et al. [13, 14], Prestridge et al. [15], Vorobieff et al. [16], and Kumar et al. [17, 18]. In these studies, a variety of diffuse density interface geometries were subjected to a planar shock acceleration at M=1.2, with gas concentration and in some cases flow field velocity data collected and analyzed. These experiments revealed a variety of interesting behaviors, including the specifics of transition to turbulence [16] and the keen dependence of the flow evolution on the initial density distribution [18]. In all cases considered, however, the initial conditions were nominally two-dimensional. As one can see from the reviewed literature, the state of the art in experimental studies of RMI flows was behind the computational work in two areas. First, there were relatively few works considering three-dimensional initial conditions (such as those produced by an oblique shock interacting with a cylindrical density interface). Second, the role of particles in RMI experiments was limited to passively following the flow as tracers for visualization, although Rightley et al. [14] mention that there is quantifiable particle lag, where particles do not follow the gas flow immediately after shock acceleration. The work described here is the first step in alleviating these shortcomings of the state of the art in experimental RMI studies. In the next chapter, a description of our experimental set up and diagnostic capabilities will be given, while the following chapter will describe the first experimental results for horizontal and oblique shock interactions with a particle-seeded heavy gas cylinder. 18

34 Chapter 2 Experimental Setup and Diagnostics 2.1 Overview Our experimental setup consists of a shock tube which is constructed from cm square 6061T 6 Aluminum tubing with a 1.27 cm wall thickness. The tube is divided into four sections; the driver, driven, test, and run-off sections, see Fig The driver section uses a circular 9.53 cm outer diameter tubing with a 1.27 cm wall thickness. The reason for using a circular driver section is simply to reduce high stress concentrations that are inherent in a square driver section. A 0.64 cm 1.22m 3.20m 0.76m 0.61m 2.6m Pressure Transducers Driver Section Driven Section Test Section Run-off Section Diaphragm Figure 2.1: Diagram showing the layout of the shock tube and the individual sections. 19

35 Chapter 2. Experimental Setup and Diagnostics diameter shaft positioned concentrically inside of the driver section has a puncturing device mounted on one end. The puncturing device consists of four mounted utility blades which puncture the diaphragm, initiating an experiment. The other end of the shaft is connected to an electronic solenoid which actuates the puncturing device. Attached to the driver section is the driven section which has two pressure transducers mounted at each end. The test sections are constructed from Lexan with a 1.27 cm wall thickness to provide transparency that allows for undistorted diagnostics of phenomena that occur during an experiment. There are multiple test sections used for injecting the SF 6 at angles of 0, 15, and 30 from vertical. The fact that the shock tube has the ability to be tilted, allows us to conduct experiments involving 3D initial conditions and oblique shock wave interactions with a dropletseeded gas cylinder. 2.2 Injection System The injection system consists of a tank of SF 6 which is fed into a chamber where it is stored temporarily during experiments. A smoke machine sitting on top of the chamber, when activated by push-button control, seeds the SF 6 with glycol droplets allowing us to visualize the gas cylinder when it is injected into the test section, see Fig. 2.2 and Fig From the chamber, the seeded mixture is fed via a Teflon tube approximately 6 mm in diameter, into a nozzle that is fixed in the top of the test section. The primary reason for using Teflon is that it is resistant to acetone which is used for fluorescence, and will not deteriorate over time. The gas cylinder exits the bottom of the test section through a hole and vents to atmosphere. After several experiments, the chamber temperature begins to rise due to the high temperature of the entering glycol droplets. This causes some turbulence in the gas cylinder which makes it difficult to extract data from the resulting images. Adding a concentric flow of air around the gas cylinder would act to minimize turbulence. 20

36 Chapter 2. Experimental Setup and Diagnostics SF 6 air air Shock Wave Figure 2.2: Diagram showing the injection system with co-flow implemented for the planar shock wave experiments. We integrated a two-phase solution by installing a co-flowing cylinder of air into our injection nozzle as well as adding ice to the chamber. This acts to cool the SF 6 as it enters the chamber and maintains an overall lower temperature inside the chamber. Our white light sources also cause the test section to rise in temperature which can cause undesirable effects, including buoyancy. With this modification we were able to increase the stability of the cylinder and mitigate the interference of turbulence in our initial conditions. 2.3 Diagnostics An Imacon 200 ultra-high speed digital camera is used to capture the shock wave interaction with the gas cylinder, see Fig. 2.4 [19]. The Imacon 200 uses multiples of intensified CCD modules providing simultaneous framing and streak recording 21

37 Chapter 2. Experimental Setup and Diagnostics SF 6 air air Shock Wave 15 Figure 2.3: Diagram showing the injection system with co-flow implemented for the oblique shock wave experiments. through a single optical axis. The camera has a maximum frame rate of 200 million frames per second from which we can capture 14 exposures at maximum resolution. High spatial and temporal accuracy make the Imacon 200 an ideal system for capturing our images. The camera is controlled via a dedicated computer with software that allows us to manipulate the way the camera performs. Illumination is provided via two Fastax floodlights originally designed for ultra-high speed cinematography. Unlike conventional floodlights, they do not exhibit brightness fluctuations with time. In addition to the floodlights, four LED arrays are use. Each LED array consists of approximately 100 individual LEDs. The camera receives a trigger from the first of two pressure transducers. The transducers are mounted in the driven section, 2.6 m apart. Analysis of the signals from the pressure transducers allows us to verify the theoretical Mach number as well as diagnose any problems that may have occurred during an experiment. A 22

38 Chapter 2. Experimental Setup and Diagnostics Figure 2.4: DRS Imaging Imacon 200 high-speed camera [19]. delay generator is also used to compensate for the time it takes the shock wave to travel the distance of the driven section into the proper location of the test section, see Fig. 2.5 [20]. 2.4 Additional Capabilities In addition to the Fastax floodlights and four LED arrays, two 532 nm (used for particle visualization) and one 266 nm (used for gas phase Planar Laser Induced Fluorescence (PLIF)) double pulsed lasers can be used for visualization purposes. The camera is oriented so that a side view and top view can be seen simultaneously in each frame see Fig Another feature of our injection system is the use of an inline filtering flask containing acetone. SF 6 is fed into the filtering flask via a glass pipette with its end submerged in the acetone. The bubbling that occurs seeds the exiting SF 6 with acetone particles that cause the gas cylinder to fluoresce under laser illumination. A combination of optical lenses, mirrors, and filters will allow us to employ planar laser 23

39 Chapter 2. Experimental Setup and Diagnostics Figure 2.5: Schematic showing how the camera, delay generator, lasers, and transducers are configured. For details refer to Chavez et al. [20]. induced fluorescence (PLIF). Manipulation of laser beams enables us to visualize a slice of the gas cylinder with which the shock wave interacts. Also, there are other features of the flow that are slightly visible in our results from the Imacon 200 that can be analyzed in more detail by using laser diagnostics in addition to other visualization techniques. Figure 2.6: Diagram showing the orientation of Imacon 200 relative to test section (left) and an individual frame representative of typical experimental results (right). 24

40 Chapter 2. Experimental Setup and Diagnostics Figure 2.7: Raw frames without acoustic pre-shock. 2.5 Acoustic Pre-shock In our preliminary experiments, we were able to capture the cause of acoustic preshock. The pre-shock was observed by earlier experimenters Kumar et al. [18]. It manifested itself as an acoustic perturbation that distorted the initial conditions prior to the arrival of the main shock wave, but its origins were not understood. With our test section attached directly to the driver section, we triggered the camera from the switch that is used to actuate the puncturing device. With the driver section pressurized, we activated the puncturing device and with two diaphragms in place we were able to see the first diaphragm get punctured and second diaphragm actually bulge and burst before the puncturing device could reach it. This late rupture of the second diaphragm is the cause of acoustic pre-shock, see Fig To avoid acoustic pre-shock, we experimented with different types of diaphragm materials. It seemed that the elasticity was too high in the diaphragm sheets that we were using. We used some more brittle sheets and this seemed to provide a more stiff rupture of the diaphragms which eliminated the acoustic pre-shock. Figs. 2.7 and 2.8 both consist of two consecutive frames showing an experiment without acoustic pre-shock as well as showing an experiment with acoustic pre-shock. The elapsed time between the frames was ms. In Fig. 2.7, the first frame 25

41 Chapter 2. Experimental Setup and Diagnostics Figure 2.8: Raw frames with acoustic pre-shock. (left) and the second frame (right) were taken at t = ms and t = ms, respectively. The puncturing device effectively ruptures both diaphragms from one frame to another. In Fig. 2.8, the first frame (left) and the second frame (right) were taken at t = ms and t = ms, respectively. The first frame shows the outer diaphragm bulging before it bursts due to yielding, not from being punctured. The additional time that it took for the diaphragms to rupture completely was due to the bulging of the second diaphragm. Acoustic pre-shock was causing problems with timing since it introduced an inconsistency into our system. By changing the diaphragm material we were able to avoid acoustic pre-shock. 26

42 Chapter 3 Results of Experiments We will present the results from experiments conducted at Mach numbers of 1.2, 1.9, and 2.4, with the shock tube at an incline of 15 from the horizontal. However, for the horizontal experiments results will only be presented for Mach numbers of 1.9 and 2.4 simply due to lack of data. We were still able to confirm earlier observations as well as reveal new features of the flow. The analysis of the oblique shots allowed us to gain information about how an oblique shock wave interacts with the gas cylinder. It was possible to analyze a rotation of the gas cylinder since some straightening occurs immediately upon the shock wave interaction. This is discussed further in section Horizontal Experiments Experimentation began with the shock tube oriented in a horizontal position, producing planar shock wave interactions with the SF 6 gas cylinder. Using ImageJ, an image analysis software, we were able to extract data such as the piston velocity, development of the counter-rotating vortex pair, as well as observe differences in these quantities when compared at different Mach numbers. The Mach 2.4 planar 27

43 Chapter 3. Results of Experiments Figure 3.1: Processed image sequence, inverted and contrast-enhanced, Mach 2.4 acceleration of SF 6 gas cylinder with shock tube in the horizontal orientation. shock wave acceleration of the gas cylinder embedded in air is shown as an example of our results, see Fig Labels in the top left corner of each frame denote time t, with t = 0 corresponding to the shock arriving at the cylinder. Each frame shows a combination of the top (above) and side (below) view. Arrows with the index a denote the location of the side view of the gas cylinder in each frame; arrows with the index b point to the corresponding top view. Note that in the first three images, the top view is completely obscured by the injection nozzle, and in the fourth image, it is partially obscured. The injection nozzle was positioned vertically in the test section. This configuration allowed us to study phenomena that are inherent with the normal acceleration of a gas cylinder composed of a heavy gas embedded in a lighter gas. Horizontal experiments allowed us to confirm observations made in earlier studies of the acceleration of an SF 6 cylinder embedded in air. The evolution of the Richtmyer-Meshkov 28

44 Chapter 3. Results of Experiments Instability can be clearly seen in both the top and side views. Once the shock accelerates the gas cylinder, the counter rotating vortex pair begins to develop. Also, glycol droplets from the gas cylinder are advected into the top and bottom boundary layers after the shock wave passes. Analysis of the pressure traces from each experiment enabled us to confirm the speed of the shock wave and verify that we were indeed reaching a particular Mach number. Further image analysis gave a piston velocity from each experiment which was verified through hand calculations using fundamental gas dynamics equations. Further analysis and experiments are planned to study the development of the boundary layer in addition to other instabilities. This, however, will require the implementation of our laser diagnostic methods both to increase the resolution of our results and allow us to take more accurate measurements. By analyzing the images from each experiment, we are able to calculate a piston velocity simply by measuring the distance in which the gas cylinder travels over the time span of the captured frames. Since we know the dimensions of our test section, we can set a scale in the image analysis software causing all of the measurements to have the particular units in which we specify. We document the frame interval of each experiment to keep track of the total amount of elapsed time. By plotting the translation of the gas cylinder versus time, a piston velocity can be calculated from the slope of the curve for a particular Mach number, see Fig There was little error between experimental piston velocities and theoretical piston velocities. A linear regression on each curve shows that the velocity of the gas cylinder is nearly constant at each Mach number after shock acceleration. For each experiment, measurements were taken from both the top and the side views. From the side view, velocity is the main parameter that is extracted. However, from the top view, measurements are made to quantify how the counter-rotating vortex pair is developing over time. Both width (in the direction normal to the streamwise) and streamwise span measurements are taken and plotted against time, see Fig

45 Chapter 3. Results of Experiments Figure 3.2: Translation of the center of mass of the gas cylinder from t = 0 ms to t = ms for Mach numbers 1.2, 1.9, and 2.4 with shock tube in horizontal orientation. and Fig When compared to the results others have obtained from similar experiments [16], our counter-rotating vortex pair develops in a very similar manner, with the geometrical feature growth consistent with hyperbolic sine-like models that were developed for fitting data from earlier studies [14]. While these trends are the same for all Mach numbers, higher values of the Mach number result in faster initial growth and slightly larger overall size. When comparing width and span measurements between various experiments, it was determined there was a small variation (±1.5 mm), likely due to initial condition fluctuations. This is represented by the error bars in the width and span plots. In the width versus time plot from 0 ms to approximately 0.05 ms, it can be seen that the development of the counter-rotating vortex pair begins at a rate that is substantially 30

46 Chapter 3. Results of Experiments Figure 3.3: Width (in the direction normal to the streamwise) of counter-rotating vortex pair versus time with shock tube in the horizontal orientation. higher than the rest of the development beyond 0.05 ms. Initially, when the shock wave interacts with the gas cylinder, much of the growth of the instability takes place toward the beginning of the total development. Beyond 0.05 ms the counter-rotating vortex pair continues to grow in width but at a much slower rate. Similar trends can be observed for the span. These effects are more dramatic, of course, at higher Mach numbers. An interesting observation when comparing the width plot from horizontal shots to the width plot from vertical shots, is that in the inclined orientation the width of the counter-rotating vortex pair develops at a much steadier and gradual rate than that of the horizontal orientation. This is discussed in the next section. 31

47 Chapter 3. Results of Experiments Figure 3.4: Span (in the streamwise direction) of counter-rotating vortex pair versus time with shock tube in the horizontal orientation. 3.2 Oblique Experiments For the experiments involving oblique shocks, the shock tube was inclined at an angle of 15 from the horizontal. This configuration creates the scenario of an oblique shock wave accelerating a heavy gas cylinder while of course the shock is still traveling normal to the longitudinal axis of the shock tube, see Fig Fig. 3.5 shows a Mach 2.4 shock acceleration of an SF 6 cylinder embedded in air. The angle between the plane of the shock and the central axis of the cylinder is 15. Each frame shows a combination of top (above) and side (below) view. The 32

48 Chapter 3. Results of Experiments Figure 3.5: Processed image sequence, inverted and contrast-enhanced, Mach 2.4 acceleration of SF 6 gas cylinder with shock tube inclined 15 from horizontal. image timings in the first twelve frames are identical to those in Fig. 3.1 with a 25 microsecond interval between all the frames. Note the advection of the cylinder material into the top boundary layer, particularly visible in the top view in the last two frames (t = ms and t = ms). The piston velocities of the oblique experiments were verified in the same way as the horizontal experiments. The rotation of the cylinder may allow us to calculate rough angular velocities (and thus circulations). We plan to study this rotation in more detail, and make correlations between the interaction of the pressure and density gradients. Experiments conducted in the inclined orientation produce results that are substantially different than those from the horizontal experiments. These differences will be discussed in more detail in the next section. The major difference, however, is in the flow evolution in the plane of the side view. When the shock wave accelerates the gas cylinder, a slight rotation, or straightening, of the cylinder occurs. The rate at 33